Tensor networks describe high-dimensional tensors as the contraction of a network (or graph) of low-dimensional tensors. Many interesting tensor can be succinctly represented in this fashion -- from many-body ground states in quantum physics to the matrix multiplication tensors in algebraic complexity. I will give a mathematical introduction to the formalism, give several examples, and sketch some of the most important results. We will discuss the role of the network, how symmetries are encoded, tensor networks as a computational model, and survey some recent algorithmic results.